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May
11
comment A congruence involving prime numbers
As long as you are in $\mathbb Z$, you even have $a^p\equiv p\pmod p$ (hence trivially $(a+b)^p\equiv a+b\equiv a^p+b^p$)
May
11
comment Conjecture: Only one Fibonacci number is the sum of two cubes
If $F_n=a^3+b^3$ then $a+b\mid F_n$. By the OPs findings, $a+b>10^5$. Heuristically, for given $d=a+b$, the probability that $d\mid F_n$ is about $\frac 1d$ so that we expect the first positive $m$ with $d\mid F_m$ at $m\approx d$. But then $F_n\ge F_m\approx \phi^m\gg m^3\approx d^3\approx a^3+b^3$. - Of course, specifically picking $d$ as a large divisor of a small $F_m$ somewhat breaks this argument ...
May
11
answered Grade School Math: Bad math, or new meanings?
May
11
comment Proving the existence of unity in $R$, where $R$ is the ring of polynomials over complex numbers with $f(0)=0$.
Is that a polynomial?
May
11
answered Improve confidence interval by averaging
May
11
answered 2014 USAMO Problem :with Points Collinear iff Sum is Constant
May
11
answered Show that $x$ is an algebraic number? Where $x$ is…
May
11
answered Does finite expectation imply bounded random variable?
May
11
comment Does finite expectation imply bounded random variable?
@mathse measure zero doesn't count ("$|X|<\infty$ a.s.")
May
10
answered An infinite non-Abelian group with an involutive automorphism that preserves only the identity?
May
10
comment Conjecture: Only one Fibonacci number is the sum of two cubes
$F_{3n}-F_{n+1}^3$ is relatively small and hence has some chance to be a cube "by chance". But otherwise the exponential growth makes the existence of larger solutions somewhat unlikely.
May
10
comment Question about the non-hausdorffness of the cofinite topology
To stress what egreg said: If $X$ is finite then the cofinite topology is the discrete topology an dhence Hausdorff. So the condition that $X$ is infinite is missing from th eproblem statement!
May
10
answered What is the minimum value of $abc$
May
10
comment What is the minimum value of $abc$
My bet: $a,b,c$ must be nonnegative integers and double root is not allowed.
May
10
comment What is the minimum value of $abc$
@Samurai I have given an explcit example with $abc=\frac5{36}$ already in my previous comment. Did you perhaps forget an important condition on $a,b,c$ (such as: nonnegative integer)?
May
10
comment What is the minimum value of $abc$
@boxed__l We cannot take $k=\pm\infty$, only consider $k\to\pm\infty$; whatever $M\in\mathbb R$ you pick, one can pick $k$ with $|k|$ large enough to ensure $k^3abc<M$. Strictly speaking, $-\infty$ is not make the minimum, but the infimum of all possible $abc$. Also, we need to ensure that there exists such a polynomial with $abc\ne0$ in the first place. $(x-\frac12)(x-\frac13)=x^2-\frac56x+\frac16$ shows this.
May
10
answered Find the sum of $\displaystyle\sum_{i=0}^k \dbinom{n+i}{i}$
May
10
comment Is a complex number whose real and imaginary parts are both transcendental transcendental?
For the second question, if $z=a+bi$ is algebraic and $b$ is algebraic, then $a=z-i\cdot b$ is also algebraic.
May
10
answered Area of a square inscribed in a triangle?
May
10
answered Why the discriminant determine whether a quadratic has real roots or not?